Conventional Boundary Integral Equation Research Articles

Stress Concentration of a Microvoid Embedded in a Bilayered Material Considering the Boundary Effects

This paper extends Eshelby’s problem of one inhomogeneity embedded in a homogeneous infinite domain to a bimaterial infinite domain. The equivalent inclusion method (EIM) was used to simulate the inhomogeneity of an inclusion with a polynomial eigenstrain. The fundamental solution of a point force in a bimaterial was used to formulate the domain integral over the inclusion. For a finite bimaterial domain, the boundary integral equation (BIE) takes into account the boundary responses by a single domain instead of utilizing the conventional multiregion BIE scheme. The EIM can be used similarly, and the elastic field can be obtained with tailorable accuracy based on the order of the polynomial eigenstrain. The algorithm is particularly suitable to simulate a defect in thin film–substrate systems or other similar bilayered materials. The stress concentration of a microvoid embedded in a bilayered solar panel was investigated. The size and location of the void relative to the interface exhibited considerable effects on the stress concentration factor. Numerical case studies demonstrated the effectiveness and accuracy of the algorithm, and parametric studies showed the boundary effects on the stress concentration of a microvoid in a finite bimaterial under a uniform far-field strain.

A Simplified Panel Method (sPM) for Hydrodynamics of Air Cushion Assisted Platforms

Air-cushion-assisted platforms (ACAPs) are floating platforms supported by both buoyancy pontoon and air cushion, which have merits of wave bending moment reduction, better stability, and hydrodynamic performance. However, there is barely a concise method that can quickly predict the motion response of ACAPs. In this paper, a simplified panel method (sPM) was presented for evaluating the hydrodynamics of ACAPs. The sPM extends the conventional boundary integral equation (BIE) to include the radiation solutions of pulsating air pressure but ignores some unimportant air-water cross terms in motion equations whose coefficients cannot be directly derived from conventional Green’s function methods. The effectiveness of the sPM was validated by experimental data from an ACAP model with one air chamber and analytical results from an oscillating water column (OWC). The numerical results demonstrate that the sPM can give desirable predictions for motion responses of the ACAP and inner pressure of the OWC as compared with results from the literature, which suggests the sPM could be approximately applied to evaluation of hydrodynamic performance of ACAPs and OWCs.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.

An isogeometric Burton-Miller method for the transmission loss optimization with application to mufflers with internal extended tubes

In this work, the problem of a muffler with internal extended inlet/outlet tubes is studied with the aim to optimize its geometric design. When boundary element method (BEM) based on the conventional boundary integral equation (CBIE) is used for structures with thin walls (internal extended tubes of small thickness), it produces nearly singular integrals which require smaller elements and higher number of Gauss quadrature points to maintain sufficient accuracy of the overall solution. An alternative approach, known as mixed or dual BEM, which consists of neglecting the tube thickness, models the wall as a single surface and prescribes BIEs of different types on each side. However, this alternative leads to some discrepancies in the obtained results compared to other numerical or experimental data especially at higher frequencies. Another approach in this work is to utilize Burton-Miller method (BM) prescribing a combination of two types of BIEs on the original thin-walled geometry. BM is introduced in the framework of isogeometric analysis (IGA) to solve the muffler problem as an interior 3D Helmholtz acoustic problem producing more accurate results. In addition, the isogeometric BM (IGA-BM) model benefits from the BEM philosophy of reducing the dimensionality of a 3D problem to consider only its surfaces/boundaries. Moreover, producing the 4-pole parameters of the transmission loss (TL) entering/exiting the inlet/outlet tubes is convenient in the framework of BEM, which easily predicts the physical variables on the problem boundary.The performance of IGA-BM model is discussed and compared with the previously published results using finite and boundary element methods, IGA models and the available experimental data. Furthermore, optimization examples using the Particle Swarm Optimization (PSO) coupled with IGA-BM are demonstrated to maximize the TL in attenuation regions by changing the internal lengths of the extended tubes.

3D isogeometric boundary element analysis and structural shape optimization for Helmholtz acoustic scattering problems

The capability of boundary element methods (BEM) for solving three-dimensional time harmonic Helmholtz acoustic scattering problems is presented in the framework of the isogeometric analysis (IGA). Both the CAD geometry and the physical boundary variables are approximated using Non-uniform Rational B-splines basis functions (NURBS) in an isogeometric setting. A detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton–Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on a smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in O(1∕r) integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles.Acoustic shape optimization in different mediums (air and water) is performed with Particle Swarm Optimization (PSO) and the results are compared with the benchmark solutions from the literature. The reference solutions are obtained with BM which deals with higher order singularities and gradient-based optimization, and requires more complicated sensitivity analysis. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement.

Frequency-Based Performance Analysis of an Array of Wave Energy Converters around a Hybrid Wind–Wave Monopile Support Structure

In this paper, we investigate, in the frequency domain, the performance (hydrodynamic behavior and power absorption) of a circular array of four semi-immersed heaving Wave Energy Converters (WECs) around a hybrid wind–wave monopile (circular cylinder). The diffraction/radiation problem is solved by deploying the conventional boundary integral equation method. Oblate-spheroidal and hemispherical-shaped WECs are considered. For each geometry, we assess the effect of the array’s net radial distance from the monopile and of the incident wave direction on the array’s performance under regular waves. The results illustrate that by placing the oblate spheroidal WECs close to the monopile, the array’s power absorption ability is enhanced in the low frequency range, while the opposite occurs for higher wave frequencies. For hemispherical-shaped WECs, the array’s power absorption ability is improved when the devices are situated close to the monopile. The action of oblique waves, with respect to the WECs’ arrangement, increases the absorbed power in the case of oblate spheroidal WECs, while these WECs show the best power absorption ability among the two examined geometries. Finally, for the most efficient array configuration, consisting of oblate spheroidal WECs situated close to the monopile, we utilize an “active” Power Take-Off (PTO) mechanism, facilitating the consideration of a variable with frequency PTO damping coefficient. By deploying this mechanism, the power absorption ability of the array is significantly enhanced under both regular and irregular waves.

Simulation of Sound Propagation Over an Infinite Impedance Plane by Using a Fast Multipole BEM

In this paper, a half-space fast multipole BEM is developed for the simulation of three-dimensional acoustic problems above an infinite impedance plane. The half-space impedance Green’s function involving a complex line source is used, so that both mass-like and spring-like impedance boundary conditions on the infinite plane can be explicitly satisfied and the infinite plane is not required to be discretized. The Burton–Miller method is employed to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Image relations of the multipole expansion coefficients are used and the half-space impedance Green’s function is modified to apply such relations to avoid calculating, translating and saving the multipole/local expansion coefficients in the image domain. An automatic integrator with adaptive interval subdivision is further adopted to calculate the line integral contained in the M2L translation formula accurately and efficiently. Numerical examples are given to show the validity and potential of the method.

Frequency-based investigation of a floating wave energy converter system with multiple flaps

In the present paper, a new Floating Wave Energy Converter System (FloWECS), consisting of a floating platform and multiple rotating flaps, is proposed and is preliminary investigated. The numerical analysis is implemented in the frequency domain under the action of regular head and oblique incident waves and it is based on a “dry” mode superposition approach. In this approach, the generalized modes concept is utilized for describing the relative to the platform rotation of the flaps, additionally to the six rigid-body modes. The required mode shapes of the generalized modes are determined through appropriate vector shape functions, which are derived in the present paper. The diffraction/radiation problem is solved using a numerical model based on the conventional boundary integral equation method. The proposed numerical formulation is, initially, validated by comparing results with the numerical ones of other investigators for the case of a single, bottom-hinged, rotating plate. Next, the exciting forces and the response of the “isolated” platform are assessed, while, finally, the hydrodynamic behaviour and the energy absorption of the FloWECS are investigated. The results illustrate, that for both examined incident wave directions, the generalized modes’ response and, thus, the system's power absorption is characterized by the existence of distinctive peaks at two different wave frequency ranges, attributed to resonance effects, as well as to the strong interaction effects between the activated flaps.

Performance of an Array of Oblate Spheroidal Heaving Wave Energy Converters in Front of a Wall

In this paper, we investigate the performance of a linear array of five semi-immersed, oblate spheroidal heaving Wave Energy Converters (WECs) in front of a bottom-mounted, finite-length, vertical wall under perpendicular to the wall regular waves. The diffraction and radiation problems are solved in the frequency domain by utilizing the conventional boundary integral equation method. Initially, to demonstrate the enhanced absorption ability of this array, we compare results with the ones corresponding to arrays of cylindrical and hemisphere-shaped WECs. Next, we investigate the effect of the array’s distance from the wall and of the length of the wall on the physical quantities describing the array’s performance. The results illustrate that the array’s placement at successively larger distances from the wall, up to three times the WECs’ radius, induces hydrodynamic interactions that improve the array’s hydrodynamic behavior, and thus its power absorption ability. An increase in the length of the wall does not lead to any significant power absorption improvement. Compared to the isolated array, the presence of the wall affects positively the array’s power absorption ability at specific frequency ranges, depending mainly on the array’s distance from the wall. Finally, characteristic diffracted wave field patterns are presented to interpret physically the occurrence of the local minima of the heave exciting forces.

Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization; 2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix; 3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

Isogeometric boundary element method for acoustic scattering by a submarine

Isogeometric analysis with the boundary element method (IGABEM) has recently gained interest. In this paper, the approximability of IGABEM on 3D acoustic scattering problems will be investigated and a new improved BeTSSi submarine will be presented as a benchmark example. Both Galerkin and collocation are considered in combination with several boundary integral equations (BIE). In addition to the conventional BIE, regularized versions of this BIE will be considered. Moreover, the hyper-singular BIE and the Burton–Miller formulation are also considered. A new adaptive integration routine is presented, and the numerical examples show the importance of the integration procedure in the boundary element method. The numerical examples also include comparison between standard BEM and IGABEM, which again verifies the higher accuracy obtained from the increased inter-element continuity of the spline basis functions. One of the main objectives in this paper is benchmarking acoustic scattering problems, and the method of manufactured solution will be used frequently in this regard.

Evaluation of the hypersingular boundary integral equation for acoustic wave-propagation simulations

We have developed a hypersingular boundary integral equation (BIE) approach to simulate wave scattering by inclusions such as karstic cavities and gypsum salts. The solution of conventional BIEs for exterior boundary value problems is of nonuniqueness and numerical instability at characteristic wavenumbers (corresponding to fictitious resonant frequencies). With the hypersingular boundary integral operator, we can handle this problem, but with the difficult-to-calculate hypersingular kernels. Our method to evaluate the singular part in the operator is more efficient than other singularity-elimination techniques. Numerical examples validate the method by comparing with the analytical solution.

Hydrodynamic performance of an array of truncated cylinders in front of a vertical wall

In the present paper, the hydrodynamic performance of a linear array of truncated circular cylinders in front of a bottom–mounted vertical wall is analysed in the frequency domain in order to investigate the effect of the wave reflections combined with the disturbances induced by the bodies themselves. Infinite and finite-length wall cases are examined and compared. The former case is formulated and solved using the matched eigenfunction expansion technique and the “direct” solution methodology for the numerical implementation. The existence of a “pure” wave-reflecting wall of infinite length is considered by utilizing the method of images. For the finite-length wall cases the conventional Boundary Integral Equation (BIE) formulation is employed. Emphasis is given on: (a) revealing the limitations of the “infinite wall” concept on the calculation of the exciting forces acting on the cylinders of the array and (b) investigating the effect of the length of the wall on the aforementioned physical quantities. The results demonstrate that the “infinite wall” concept leads to under- or over-estimation of the heave exciting forces at small wave numbers, while this concept seems not to be adequate for the accurate calculation of the horizontal exciting forces in cases of small outer cylinder's center–to–wall's edge distance. The change of the wall's length affects the surge and the heave exciting forces, while, the presence of the wall, contrary to the case of the isolated array, leads to an intense variation pattern of the cylinders' sway exciting forces. This pattern is characterized by successive local minima, which can be related to the existence of a symmetrical wave field in the seaward and leeward sides of the cylinders.

Desingularization of matrix equations employing hypersingular integrals in boundary element methods using double nodes

In boundary element methods, using double nodes at corners is a useful approach to uniquely define the normal direction of boundary elements. However, matrix equations constructed by conventional boundary integral equations (CBIEs) become singular under certain combinations of double node boundary conditions. In this paper, we analyze the singular conditions of the CBIE formulation for cases where the boundary conditions at the double node are imposed by combinations of Dirichlet, Neumann, Robin, and interface conditions. To address this singularity, we propose the use of hypersingular integral equations (HBIEs) for wave propagation problems that obey the Helmholtz equation. To demonstrate the applicability of HBIE, we compare three types of simultaneous equations: (i) CBIE, (ii) partial-HBIE where the HBIE is only applied to the double nodes at corners while the CBIE is applied to the other nodes, and (iii) full-HBIE where the HBIE is applied to all nodes. Based on our numerical results, we observe the following results. The singularity of the matrix equations for problems with any combination of boundary conditions can be resolved by both full-HBIEs and partial-HBIEs, and the partial-HBIE exhibits better accuracy than the full-HBIE. Furthermore, the computational cost of partial-HBIEs is smaller than that of full-HBIEs.

Evaluation of hypersingular and nearly singular integrals in the Isogeometric Boundary Element Method for acoustics

Special integration routines for hypersingular and nearly singular integrals that arise in an Isogeometric Boundary Element Method (IGABEM) are presented. The IGABEM is applied to an acoustic problem in the frequency domain that corresponds to the Helmholtz equation in three dimensions. A major topic of such an acoustical BEM is the non-uniqueness of the conventional boundary integral equation, which can be corrected by the Burton–Miller formulation at the cost of a hypersingular integral that requires the presented routines. The combination of the BEM with an Isogeometric Analysis (IGA) is advantageous, since the exact CAD description is directly incorporated to the numerical simulation. As the geometry is represented exactly, the integration routines become even more relevant and cannot be masked by the geometrical approximation. For the conventional approximation with plane Lagrange elements, a variety of different integration routines exist, but are not investigated in the scope of an IGA. Therefore, the hypersingular integration scheme of Guiggiani is adapted to be applicable to this new methodology. A direct evaluation of the hypersingular integral is possible, instead of an integration over the complete surface, as in the often used regularized boundary integral equation. Additionally, the sinh-transformation for nearly singular integrals is introduced to achieve more accurate results with the same number of integration points as the conventional Gaussian integration. The new formulation is analyzed under the special case of the oscillatory kernels in acoustics.

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation (CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation (BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method. With this method, for the Neumann boundary value problem (BVP) of an interior region, a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.

用于解析函数复分析的共轭边界元法

An analytic function is composed of 2 real conjugate harmonic functions, of which the complex analysis plays an important role in the fields of applied mathematics and mechanics. A set of weighted residual equations were proposed and proved to be equivalent to the approximate solution to the original problem involving 2 governing equations in the domain, the boundary condition and the CauchyRiemann equation at the boundary. 2 conventional direct boundary integral equations at the boundary collocation points were deduced from 2 of the weighted residual equations, and 1 finite difference equation was deduced from the rest one. The mathematical problem arising from the illconditioned linear equations was solved and the Cauchy integral equation was adopted for numerical calculation of the fields at the internal points inside the domain. Finally, the proposed conjugate boundary element method with constant elements was completely established. 3 examples demonstrate that, the proposed method is valid for analytic functions in terms of the power function, the exponential function and the logarithmic function in interior or exterior domains, and the error estimation of the proposed method is at the same order as that of the boundary element method for 2D potential problems.

A robust and accurate formulation of molecular and colloidal electrostatics.

This paper presents a re-formulation of the boundary integral method for the Debye-Hückel model of molecular and colloidal electrostatics that removes the mathematical singularities that have to date been accepted as an intrinsic part of the conventional boundary integral equation method. The essence of the present boundary regularized integral equation formulation consists of subtracting a known solution from the conventional boundary integral method in such a way as to cancel out the singularities associated with the Green's function. This approach better reflects the non-singular physical behavior of the systems on boundaries with the benefits of the following: (i) the surface integrals can be evaluated accurately using quadrature without any need to devise special numerical integration procedures, (ii) being able to use quadratic or spline function surface elements to represent the surface more accurately and the variation of the functions within each element is represented to a consistent level of precision by appropriate interpolation functions, (iii) being able to calculate electric fields, even at boundaries, accurately and directly from the potential without having to solve hypersingular integral equations and this imparts high precision in calculating the Maxwell stress tensor and consequently, intermolecular or colloidal forces, (iv) a reliable way to handle geometric configurations in which different parts of the boundary can be very close together without being affected by numerical instabilities, therefore potentials, fields, and forces between surfaces can be found accurately at surface separations down to near contact, and (v) having the simplicity of a formulation that does not require complex algorithms to handle singularities will result in significant savings in coding effort and in the reduction of opportunities for coding errors. These advantages are illustrated using examples drawn from molecular and colloidal electrostatics.

The Calculation of Potential Derivatives by Using NBIE for Anisotropic Potential Problems

AbstractThe natural boundary integral equation (NBIE) is developed to calculate potential derivatives for potential problems with anisotropic media. Firstly, the governing equation of the two-dimensional anisotropic potential problem is transformed into standard Laplace equation by a coordinate transformation method. Then a potential derivative boundary integral equation named as NBIE is extended to solve the anisotropic potential problem. The most important virtue of the NBIE is that the singularity of the integral kernel function is reduced by one order in comparison with the conventional potential derivative boundary integral equation(CDBIE). Therefore the new potential derivative boundary integral equation only contains strongly singular integrals rather than hyper-singular integrals. Thus the NBIE can calculate more accurate potential derivative results for both boundary nodes and interior points. Moreover, in combination with the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points very close to the boundary than the CDBIE. Numerical examples on heat conduction in anisotropic media demonstrate the accuracy and efficiency of the NBIE.

Symmetric Complex Boundary Element Scheme for 2-D Stokes Mixed Boundary Value Problem

For 2-D Stokes mixed boundary value problems we construct a boundary<br />integral equation which couples a conventional boundary integral equation<br />for the velocity with a hypersingular boundary integral equation for the<br />traction. Expressing terms in the equation by complex variables, we obtain a<br />complex boundary integral equation and realize symmetrization of boundary<br />element scheme by Galerkin method. Applying a boundary limit method, we<br />obtain exact calculation formulae for calculation of hypersingular boundary<br />integrals. It is shown that all divergent terms in hypersingular integrals<br />cancel each other out.